Synchronization of unified chaotic system based on passive control

Physica D 225 (2007) 55–60 www.elsevier.com/locate/physd

Synchronization of unified chaotic system based on passive control Faqiang Wang ∗ , Chongxin Liu School of Electrical Engineering, Xi’an Jiaotong University, Xi’an 710049, China Received 4 March 2006; received in revised form 4 July 2006; accepted 29 September 2006 Available online 15 November 2006 Communicated by R. Roy

Abstract Chaos synchronization in the unified chaotic system was discussed using passive control. On the basis of the property of the passive system, the passive controller is designed and synchronization of two identical unified chaotic systems with different initial conditions is realized. Simulation results for two Lorenz, two L¨u and two Chen chaotic systems are provided to show the effectiveness of the proposed synchronization method. c 2006 Elsevier B.V. All rights reserved.

Keywords: Unified chaotic system; Passive synchronization control; Asymptotically stabilized

1. Introduction Inspired by the pioneering work of Pecora et al. in 1990 [1], chaos synchronization has recently attracted great attention due to its potential applications for secure communication [2]. Previously, many different techniques and methods have been proposed for achieving chaos synchronization, such as bi-directional coupling [3], nonlinear control [4], adaptive control [5], backstepping control [6], impulse control [7], active control [8], and so on. On the other hand, the concept of passivity of nonlinear systems attracted new interest in nonlinear system control. For example, Wen [9] applied this technique to design the controller whose structure is of linear feedback form to control the Lorenz system to zero and any desired equilibria. Qi et al. [10] also applied this technique to design a controller to control a unified chaotic system to zero and any desired equilibrium. In this paper, on the basis of the property of the passive system, synchronization of a unified system [11] which contains the Lorenz [12] and Chen [13] systems as two extremes and the L¨u system [14] as a special case is studied. The essential conditions under which the error dynamical system could be equivalent to a passive system and globally asymptotically stabilized at zero equilibrium points via smooth state feedback are derived. ∗ Corresponding address: Xi’an Jiaotong University, School of Electrical Engineering, P.O. Box 2302, Xi’an 710049, China. Tel.: +86 29 82675590. E-mail address: [email protected] (F. Wang).

c 2006 Elsevier B.V. All rights reserved. 0167-2789/$ - see front matter doi:10.1016/j.physd.2006.09.038

This paper is organized as follows. In Section 2, on the basis of the character of the passive system, a rather simple conclusion of a control error dynamical system is derived for stabilizing at zero equilibrium points through a feedback control. In Section 3, a brief description of the unified system is introduced and the approach is applied to this system. In Section 4, numerical simulations are given for illustration and verification. Finally, some concluding remarks and comments are given. 2. The theory of passive control Consider the following differential equation: 

x˙ = f (x) + g(x)u y = h(x).

(1)

Here x ∈ X ⊂ Rn is the state variable. f (x) and g(x) are smooth vector fields. u(t) ∈ U is the input. f (0) = 0 and h(x) is a smooth mapping. Definition 1. The system (1) is a minimum phase system if L g h(0) is nonsingular and x = 0 is one of asymptotically stabilized equilibrium points of f (x). Definition 2. The system (1) is passive if the following two conditions are satisfied. 1. f (x) and g(x) exist and are smooth vector fields; h(x) is also a smooth mapping.

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2. For any t ≥ 0, there is a real value β that satisfies the inequality Z t u T (τ )y(τ )dτ ≥ β (2) 0

or there are real values β and ρ > 0 that satisfy the inequality Z t Z t T ρy T (τ )y(τ )dτ. (3) u (τ )y(τ )dτ + β ≥ 0

0

Definition 3. The system (1) is passive if there is a continuous differentiable nonnegative function: X → R, V (0) = 0, named the storage function, or continuous positive definite storage functions V (x) and S(x), such that ∀x ∈ X Z t y T (s)u(s)ds (4) V (x) − V (x0 ) ≤ 0

V (x) − V (x0 ) =

t

Z 0

y T (s)u(s)ds −

t

Z

S(x(s))ds.

(5)

0

When we let z = ϑ(x), the system (1) will be changed into the following generalized form:  z˙ = f 0 (z) + p(z, y)y (6) y˙ = b(z, y) + a(z, y)u where a(z, y) is nonsingular for any (z, y). Theorem 1. If the system (1) has relative degree [1, 1, . . .] at x = 0 and system (1) is a minimum phase system, the system (6) will be equivalent to a passive system and asymptotically stabilized at equilibrium points through the local feedback controller as follows:   ∂ u = a(z, y)−1 −bT (z, y) − W (z) p(z, y) − α0 y + v (7) ∂z where W (z) is the Lyapunov function of f 0 (z), α0 is a positive real value and v is an external signal which is connected to the reference input. Proof. Suppose that 1 2 y 2 d ∂ W (z) V (z, y) = z˙ + y y˙ dt ∂z ∂ W (z) ∂ W (z) = f 0 (z) + p(z, y)y ∂z ∂z + yb(z, y) + ya(z, y)u.

V (z, y) = W (z) +

(8)

(9)

Because system (1) is a minimum phase system, the inequality can be obtained as ∂ W (z) f 0 (z) ≤ 0. ∂z

(10)

So d ∂ W (z) V (z, y) ≤ p(z, y)y + yb(z, y) + ya(z, y)u. dt ∂z

(11)

Taking Eq. (7) into inequality (11), the above inequality can be rewritten as d V (z, y) ≤ −αy 2 + vy. (12) dt Then, integrating over both sides of (12), Z t Z t 2 v(τ )y(τ )dτ. (13) αy(τ ) dτ + V (z, y) − V (z 0 , y0 ) ≤ − 0

0

For V (z, y) ≥ 0, let V (z 0 , y0 ) = µ, the above inequality can be rewritten as Z t Z t α0 y(τ )2 dτ + V (z, y) v(τ )y(τ )dτ + µ ≥ 0

0

t

Z ≥

α0 y(τ )2 dτ.

(14)

0

According to Definition 2, the system (6) is a passive system. Because W (z) is radially unbounded, it follows from Eq. (8) that V (z, y) is also radially unbounded, so that the closed-loop system is bounded state stable at [z T , y]T . This means that we can use the local feedback control which is described by form (7) to regulate the system (6) to the equilibrium points and Theorem 1 is proved.  3. Synchronization of the unified chaotic system The nonlinear differential equations that describe the unified chaotic system, as a driving system, are  x˙1 = (25α + 10)(x2 − x1 ) x˙2 = (28 − 35α)x1 − x1 x3 + (29α − 1)x2 (15)  x˙3 = x1 x2 − (8 + α)x3 /3 where x1 , x2 , x3 are state variables and α ∈ [0, 1] is a system parameter. The system is chaotic for any α ∈ [0, 1]. When 0 ≤ α < 0.8, system (15) is called the generalized Lorenz chaotic system. When α = 0.8, system (15) is called a L¨u chaotic system. When 0.8 < α ≤ 1, system (15) is called the generalized Chen chaotic system. Examples of the Lorenz chaotic attractor, the L¨u chaotic attractor, the Chen chaotic attractor are obtained, as depicted in Figs. 1–3, respectively. The response system which has the controller added into the second formula of the system (15) is given by   y˙1 = (25α + 10)(y2 − y1 ) y˙2 = (28 − 35α)y1 − y1 y3 + (29α − 1)y2 + u (16)  y˙3 = y1 y2 − (8 + α)y3 /3. Let e1 = y1 − x1 , e2 = y2 − x2 , e3 = y3 − x3 . The error dynamical system between the driving system (15) and response system (16) can be expressed by  e˙1 = (25α + 10)(e2 − e1 ) e˙2 = (28 − 35α)e1 − y1 y3 + x1 x3 + (29α − 1)e2 + u (17)  e˙3 = y1 y2 − x1 x2 − (8 + α)e3 /3. This is for  y1 y3 − x1 x3 = e1 e3 + x1 e3 + x3 e1 y1 y2 − x1 x2 = e1 e2 + x1 e2 + x2 e1 .

(18)

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Fig. 1. Lorenz chaotic attractor.

Fig. 2. L¨u chaotic attractor.

Fig. 4. The maximum of x2 versus α.

Fig. 3. Chen chaotic attractor.

So, system (17) can be changed into the following form:  e˙1 = (25α + 10)(e2 − e1 )    e˙2 = (28 − 35α)e1 − (e1 e3 + x1 e3 + x3 e1 ) (19) + (29α − 1)e2 + u    e˙3 = e1 e2 + x1 e2 + x2 e1 − (8 + α)e3 /3.

The aim is to design the controller u for stabilizing the error dynamical system (19) at zero equilibrium points. Each chaotic system has its bounds; the following inequality can be obtained within α ∈ [0, 1]: |x2 | ≤ x2 max = M2 = 34.6.

(20)

The relationship between the maximum of x2 , namely M2 , and system parameter α is shown in Fig. 4.

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Fig. 5. Synchronization of two identical Lorenz chaotic systems with each other. (a) Time waveforms of the states x1 (solid) and y1 (dashed), (b) time waveforms of the states x2 (solid) and y2 (dashed), (c) time waveforms of the states x3 (solid) and y3 (dashed), (d) time waveform of the norm kek of the error vector.

! x22 8+α − − z2. 3 4M2 (25α + 10) 2

Theorem 2. We choose the controller as follows: u = (x3 − M2 (25α + 10) − 28 + 35α)e1 + (α0 + 1 − 29α)e2 .

(21)

The error dynamical system (19) will be asymptotically stabilized at zero equilibrium points and the two unified chaotic systems with different initial conditions will be synchronized. Proof. Let z 1 = e1 , z 2 = e3 , y = e2 ; the system (19) can be changed into the following form:  z˙ 1 = (25α + 10)(y − z 1 )    z˙ 2 = z 1 y + x1 y + x2 z 1 − (8 + α)z 2 /3 (22) y˙ = (28 − 35α)z 1 − (z 1 z 2 + x1 z 2 + x3 z 1 )    + (29α − 1)y + u. So  f 0 (z) = [−(25α + 10)z 1 x2 z 1 − (8 + α)z 2 /3]T      p(z, y) = [25α + 10z 1 + x1 ]T b(z, y) = (28 − 35α)z 1 − (z 1 z 2 + x1 z 2 + x3 z 1 )    + (29α − 1)y   a(z, y) = 1.

(23)

M2 2 1 2 z + z . 2 1 2 2 Then we have W (z) =

8+α 2 W˙ (z) = −M2 (25α + 10)z 12 + x2 z 1 z 2 − z 3 2  2 x2 z 2 = −M2 (25α + 10) z 1 − 2M2 (25α + 10) x22 8+α 2 z2 − z 4M2 (25α + 10) 2 3 2 2  x2 z 2 = −M2 (25α + 10) z 1 − 2M2 (25α + 10) +

This is for x22 M22 8+α 8+α − ≥ − 3 4M2 (25α + 10) 3 4M2 (25α + 10) 4(α + 8)(25α + 10) − 3M2 = 12(25α + 10) > 0. (26) So W˙ (z) < 0.

(27)

So, W (z) is the Lyapunov function of f 0 (z) and f 0 (z) is globally asymptotically stable. Meanwhile, L g h(0) is nonsingular; according to the Definition 1, the system (22) is a minimum system. On the other hand, the system (22) has relative degree [1, 1, . . .] at x = 0. So, according to Theorem 1, the feedback controller can be derived: u = (x3 − M2 (25α + 10) − 28 + 35α)z 1 + (α0 + 1 − 29α)y + v,

We choose

(25)

(28)

namely (24)

u = (x3 − M2 (25α + 10) − 28 + 35α)e1 + (α0 + 1 − 29α)e2 + v

(29)

where α0 is a real constant. When we let v = 0, the error dynamical system (19) will be asymptotically stabilized at the zero equilibrium point and the two unified chaotic systems with different initial conditions will be synchronized and Theorem 2 is proved.  4. Simulation results In this section, the fourth-order Runge–Kutta method is used to solve the system of differential equations (15) and (16) with

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Fig. 6. Synchronization of two identical L¨u chaotic systems with each other. (a) Time waveforms of the states x1 (solid) and y1 (dashed), (b) time waveforms of the states x2 (solid) and y2 (dashed), (c) time waveforms of the states x3 (solid) and y3 (dashed), (d) time waveform of the norm kek of the error vector.

Fig. 7. Synchronization of two identical Chen chaotic systems with each other. (a) Time waveforms of the states x1 (solid) and y1 (dashed), (b) time waveforms of the states x2 (solid) and y2 (dashed), (c) time waveforms of the states x3 (solid) and y3 (dashed), (d) time waveform of the norm kek of the error vector.

step size equal to 0.001 in all numerical simulations. Choose the value of α0 equal to 0.01 and initial conditions of the driving system (15): x1 (0) = −1, x2 (0) = −1, x3 (0) = 1, and of the response system (16): y1 (0) = 4, y2 (0) = −4, y3 (0) = 4. After 25 s, the controller is added to the response system. When α = 0, M2 equals 25.2 and system (15) is a

Lorenz chaotic system. When α = 0.8, M2 equals 34.2 and system (15) is a L¨u chaotic system, and when α = 1, M2 equals 34.6 and system (15) is a Chen chaotic system. The simulation results for synchronization of the Lorenz, L¨u and Chen chaotic systems are shown in Figs. 5–7, respectively. As expected, one can observe that the trajectories of the response

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system asymptotically approach those of the driving system as illustrated in Figs. 5(a)–(d), 6(a)–(d), and 7(a)–(d), and the norm kek of the error vector eventually converges to zero after the controller is activated as shown in Figs. 5(e), 6(e), and 7(e), respectively. These results of the simulation imply that the two unified chaotic systems are synchronized with each other and validate the effectiveness of the proposed method. 5. Conclusions On the basis of the property of the passive system, a feedback controller for synchronization of two unified chaotic systems with different initial conditions is derived. The effectiveness of this proposed synchronization method has been validated by numerical simulation results for Lorenz, L¨u and Chen chaotic systems. In addition, this method can be applied to synchronization of two different chaotic systems with different initial conditions. References [1] L.M. Pecora, et al., Synchronization in chaotic systems, Phys. Rev. Lett. 64 (1990) 821–824. [2] P. Colet, R. Roy, Digital communication with synchronized chaotic lasers,

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